The present invention relates to a new and distinct variety of peach tree named ‘NJ359’. Our new tree resulted from crossing ‘Flameprince’ (non-patented) as the seed parent with our proprietary peach seedling selection ‘NJ260’ (U.S. Plant Pat. No. 4,572) as the pollen parent. The new variety differs from seed parent ‘Flameprince’ in that the new variety produces slightly larger fruit and that has better flavor and aromatics. The new variety differs from pollen parent ‘NJ260’ in that the new variety produces more attractive fruit that have superior eating quality. The resulting tree was selected when growing in a cultivated area as the 76th tree in the 65th row of Block K at a fruit research farm in Cream Ridge, N.J.